Question

natics
х
(9) A triangle is drawn by joining a point on a semicircle to the ends of
the diameter. Then semicircles are drawn with the other two sides as
diameter.
Prove that the sum of the areas of the blue and red crescents in the
second picture is equal to the area of the triangle.​

Answer 1

Answer

From, the given diagram,

∠ACB = 90° (since, angle on a semicircle is right angle)

Thus, by Pythagoras theorem

AC2 + CB2 = AB2

⇒ a2 = b2 + c2 …..(1)

where, r = radius

Here,

Now, by second diagram

We get

where, r = radius

Here,

where, r = radius

Here,

Thus,

Sum of Area of crescent = S2 + S3 + area of Δ ABC – S1

(from(1))

⇒ Sum of Area of crescent = area of Δ ABC